MA 123 Elementary Calculus and its Applications
Paper Homework 7-10
Due: 21 July 2008
Instructions
Answer the question in each story. Show your work and explain your solution in
English ā you must have lots of words written down for each problem to recieve
credit. This means saying what your are doing at each step and why you are
doing it. (This is for your benifit. Explaining in writing how you solved the
problem forces you to think about the process and admit to yourself when you
donāt understand what you are doing.)
7. A Piston Consider an (idealized) piston that moves up and down in a
cylindrical sleeve so that the air column inside has a fixed mass of 1kg
compressed into a cylinder with a variable height hand radius .5m. The
height hof the air column starts at 1.1mand varies between that and
0.1min such a way that
dh
dt =āĻ
2sin īĻtsā1īm
s
where tis the time elapsed in seconds. (The sā1is a unit needed to make
the units work our right in this formula. The unit sā1is also called a
hertz, with symbol Hz . This may be familiar from processor clock speeds,
sound frequencies, or radio frequencies.) The density of the air is
Ļ=M
V
meaured in kg/m3, where M= 1kg is the mass of the air column and V
is the volume of the air column. (Ļis the lowercase greek letter ārhoā.)
Question: At what rate is the denisty of the air is increasing when the
height of the air column is 0.25mand t= 3.25318331111. Give an approx-
imate answer rounded to the nearest 0.1kg
m3s(neqarest tenth of a kilogram
per cubic meter per second).
Warning: You must be sure that your calculator is in āradiansā mode.
Hit the MODE button and make sure that āradiansā and not ādegreesā
is selected.
8. Book in a Box Sally is making a custom sized rectangular scrapbook to
give her sister as a gift. She wants to put the book in an elliptical gift box.
The box is 8 inches wide and 12 inches long. The graph of the equation
īx
aī2
+īy
bī2
= 1
is an ellipse centered at the origin that has length 2aand width 2b.
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